A theorem on semi-continuous set-valued fucntions

A THEOREM ON SEMI-CONTINUOUS SET-VALUED FUNCTIONS

BY E. MICHAEL 1. Introduction. The purpose of this paper is to prove a rather curious result bout semi-continuous set-valued functions (Theorem 1.l), and to derive some applications to open and closed point-valued functions. Recall that if X and Y are topological spces, nd if 2 r denotes the spce of non-empty subsets of Y, then function X 2 r is clled lower (respectively

:

upper) semi-continuous if

-

u (respectively {x xi(x) /x Xl(z) c u}) is open in X for every open U C Y. An important example is the ease where u-1 for some function u from Y onto X; in this case, is is of the form lower (respectively upper) semi-continuous if and only if u is open (respectively closed). THEOREM 1.1. Let X be paracompact, Y a metric space, and : X 2 lower semi-continuous with cache(x) complete. Then there exist b X 2 " and X 2

- --

such that

(a) (x) C O(x) C (x) for all x e X, (b) (x) and O(x) are compact for all x X, (c) is lower semi-continuous, (d) is upper semi-continuous. COROLLARY 1.2. Let E be a metric space, F paracompact (Hausdorff will in (b)), and f: E.---> F open and onto with f-l(y) complete for every y F. Then () There exist subsets E" E’ E such that f(E") f(E’) F, fiE’ is closed, f E" is open, and for each y F the sets (f E’)-(Y) and (f E")-(Y)

suce

_

are compact.

(b) If B F is compact, then there exists a compact A E such that f(A) B. (c) If f is continuous, then F is metrizable. Observe that Corollary 1.2. (b) generalizes result of Bourbaki [1, 2, Proposition 18], where is continuous, and E itselfrather than just the sets ]-(y) is required to be complete. It should be noted that the roles of open and closed maps in Corollary 1.2 () cannot be reversed. For instance, the map f: [0,3] [0,2], defined by f(x) x x_ 3, is conlif 1 <: x x- lif2 if0 x _< 1, f(x) 2, andf(x) tinuous and closed, but there exists no A C [0, 3] such that f(A) [0, 2] and f lA is open. The proof of Theorem 1.1 will be found in 2 and 3; the lemm in 2 my

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:Received December 16, 1958. This work was supported by the Office of Naval Research under Contract Nonr 477(21), Project Number NR 043 207.

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648

E. MICHAEL

hold some independent interest, and will also be used in [4]. The short proof of Corollary 1.2 is contained in 4, and 5 gives an example showing why we need completeness for our results.

A lemma. LEMM 2.1. Let X be paracompact, Y a topological space, X --> 2 r lower semi-continuous, and {(},.1 a sequence of continuous pseudometrics on Y. Then for each integer n > 0 there exists an index set A a locally finite open covering with a and a map {U} ,e of X, elements y.(x) (x) whenever x such are all the conditions that following onto, satisfied for n. + then .(y(x), y(x’)) < (a) U a e and x, x’ (b) If a & then U. W U :(a)}. then z.(ya(x), y.(x)) < 1. (c) If a e :(a), and x Proof. By induction. For convenience we start with n 0, after defining the pseudometric z0 by z0(y, y) 0 for all y, y, Y. We can therefore begin by letting Ao be a one-element set, say {v}, letting U. X, and for each x X letting y(x) be any element of (x). Suppose that we have everything up to n, and let us construct it for n + 1. For Let .a Without loss of generality, we may assume that + 2.

..

a

each x

,,

(1)

let

Then each V.(x) is open in

.

.

..

{x’ u

Y.(x)

"

q+(y.(x), (x’))

.. .

< }.

by the lower semi-continuity of

then is a relatively open covering of a locally finite, relatively open refinement

,

and if we let

.

has is paracompact, which in turn has a relatively open refinement {W},(.) such that B(a). C W for all We may assume that the family of index sets {B(a)}.. is disjoint, so we let and we define

.+

A+,

Since

{W},(,)

W

.B(a),

A by (B)

a

if

Note that :(a) B(a) for 11 a u A. Now let A.+. We must define the open set U, and the points y(x) for Denotev()bya. Let Ua W U. Pickanxesuchthat U C V.(x), and then for each x U use (1) to pick y(x) (x) such that (2) z(y(x), y.(x)) < Let us verify our requirements. Since U [ r:(a) is clearly a locally finite (with respect to X) open covering of U. for all a A, it follows, by our in-

x.

.

649

SEMI-CONTINUOUS SET-VAIUED FUNCTIONS

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ductive assumption on {U,} a,An, that Ua},A.+I is a locally finite open covering of X. It remains to check conditions (a) (c). The validity of (b) is clear, and that of (a) (with n replaced by n -{- 1) follows from (2). As for (c), note that if a A,/ -’(a), and x then n(y(x), y,(X)) <__ O’,,(y(x), y (X)) q- .(y (X), y (X)) < 1/2 q1. This completes the proof. 3. Proof of Theorem 1.1. Begin by applying Lemma 2.1 with cn 2p, where p is the metric on Y, and let A, v, U, and y,(x) be as in Lemma 2.1. Now let

A

{

and for each x

{a}=l

lAm

and n(+,)

,,

for all n},

X, let A(x)

(x)

{eAixeU. for all n}, {Alxe/. forall n}.

Note first that if x X and a is in A(x) or (x), then y.(x) is defined for all n, and {y(x)}n., is a Cauchy sequence (by 2.1 (c)). Since (x) is complete, this sequence must have a limit, which we denote by y,(x). We now let o(X) Oo(X)

{y.(x) {y.(x)

A(x)}, ae

_(x)},

and we let

(z) o(x)

(co(z))-, (Co(X))-.

for every x X. Before proving that the above and satisfy our conditions, let us make some observations. (1) If a _(x), then p(y,,(x), y.(x)) < 2 -(n-’. (By 2.1 (c)). (2) If a A(x), a’ %_(x’), and o an then p(y,(x), y,(x’)) < 2-(’(This follows from (1) and 2.1 (a)). (By (3) If A and x, Ua, then there exists an , A(x) such that o

,

2.1 (b)).

,

..

then there exists an e (x) such that o (This also follows from 2.1 (b) which, together with local finiteness, implies that

(4) If

A and x

It follows from (3) that A(x) is never empty, nd henee neither are k(x) or a(x). It remains to cheek requirements (a) (d). (a) This is obvious from the definitions. (b) It suffices to observe that, by (1) and loeal finiteness, Oo(x)nd hence certainly ffo(x)is totally bounded for all x. (e) It is clearly sufficient to show that ko is lower semi-continuous. So given x e X, y e /o(x), and e > 0, we must find a neighborhood W of x such that

.

.’o(X’) intersects S,(y) for all x’ W (where S,(y) denotes the open e-sphere about y). Note first that y y,(x) for some a e A(x). Now pick an n such that 2 -(’-a) < e, and let W U,.. If now x’ W, then by (3)there exists an a But then y,,(x’) o(X), and p(y,,(x’), A(x’) such that a y,(x)) < by (2), which is what we needed. (d) We must show that if x X and V is a neighborhood of O(x) in Y, then there exists a neighborhood W of x in X such that O(x’) C V for all x’ W. Now since (x) is compact, ,we can find > 0 such that S(O(x)) C V (where S,(E) denotes/.JS,(y)). It therefore suffices to pick W such that Oo(X ) C Si,(Oo(x)) for every x’ W. To do this pick an n such that 2 -(-a) 1/2e, and let

. ,,

w x-

,,. .

Suppose that x’ W and y Oo(X’). Now y y,(x’) for some a’ (x’). Then x’ hence x e and thus by (4) there exists an _(x) such that a, a: But then y,(x) Oo(X), and p(y,(x), y,,(x’)) <: 1/2 by (2), which is all we had to show. This completes the proof of the theorem. 3. Proof of Corollary 1.2. (a) Define :F --+ 2 by (y) u -l(y). By the remark preceding Theorem 1.1, this is lower semi-continuous, and thus satisfies all the requirements of Theorem 1.1. With and 0 as in Theorem 1.1, let E’ k.),,.O(y) and E" L),Fk(y). Again by the remark preceding Theorem 1.1, E’ and E" satisfy all our requirements. (b) Applying (a) to the function g f f-(B), we obtain a set A C f-l(B) such that g(A) B, g A is closed, and (g A)-(Y) is compact for every y in the compact set B. But this implies (see, for instance, [2, Theorem 1]) that A is compact. (c) We need only apply to f E’ (where E’ is as in (a)) the theorem (see S. Hanai and K. Morita [3, Theorem 1] or A. H. Stone [5, Theorem 1]) which asserts that the continuous closed image of a metrizable space is metrizable if (and only if) inverse images of points have compact boundaries. This comletes the proof. 4. An example. In this section we construct an exampleto show that 1.2(b) (and hence 1.2(a) and 1.1) can become false if one fails to assume completeness. Example 4.1. Let be the natural projection from the unit square S onto the unit interval F. Let E be a space obtained from S by removing from each E. Then f is open (and vertical interval .--l(y) a point x and let f continuous), but the sets f-(y) are not complete. Let us show that the points x can be picked so that there exists no compact A C E with f(A) F. Let a {A C S A is compact, r(A) F}. Then card a 2 (since S has a countable base), and hence there exists a function from F onto a. For every y F we now let x be any element of (y) (% r-l(y). This works, for if A C S is compact with (A) F, then A (y) for some y F, whence x,, A, and hence A ([: E.

.

SEMI- CONTINUOUS SET-VALUED FUNCTIONS

651

REFERENCES 1. N. BOURBAKI, Topologie G’n$rale, (haptr 9, 2nd edition, Hermann, 1958. 2. E. HALFAR, Compact mappings Proceedings of the American Mathematical Society, vol. 8(1957), pp. 828-830. 3. S. HANAI AND K. MORITA, Closed mappings and meric space, Proceedings of the Japan Academy, vol. 32(1956), pp. 10-14. 4. E. MICHAEL, Con16 structures and continuous selections, Canadian Journal of Mathematics, vol. 11 (1959), pp. 556-575. 5. A. H. STON., Merizablity of decomposigon spaces, Proceedings of the American Mathematical Society, vol. 7(1956), pp. 690-700.

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